5.3 Kohonen’sSelf-Organizing Network Topology knownasKSONor self-organizing map (SOM) belongs to class of unsupervised learning networks updates its weighting parameters without the … .. need for a per- formancefeedback from a teacher or network trainer Major feature-nodes distribute themselves across input space to recognize groups of similar input vectors, while output nodes compete among themselves to be fired one at a time in response to a particular input vector-process known as competitive learning-winner-take-all …

2 Fundamentals of Fuzzy Logic Systems i. A and Bareequaliff their membership functions are identi- cal. ii. A’scompliment, written as A0 is defined thusf A0 =1f A . iii. Ais contained in B (a.k.a. subset, smaller than) iff f A f B 2.1 FuzzyLogic Operations i. A[B=f A _f B ii. A\B=f A ^f B iii. AB=f A f B A\B iv. A+B=f A+B =f A +f B only meaningful when<1 v. A B= (A0 B0)0=A+BAB vi. jA Bj=f jA Bj =jf A f B j i. Boundary conditions, C () = X;C (X) = ii. Non-increasing. iii. Involutive: C (C (A)) =A T-Norm-Generalized Intersection TheT-normisnon-decreasing in each argument, commutative, associative, and satisfies the boundary condition aT 0=0. AnS-norm is the opposite. A\B (x) =T ( A (x) ; B (x)) S-Norm-T-Conorm, Generalized Union TheS-normisnon-decreasing in each argument, commutative, associative, and satisfies the boundary condition aS 0=a;aS 1=1. Set Inclusion In the context of fuzzy logic, a set maybe partially included in another set. In that case it is convenient to define a grade of inclusion of a fuzzy setA in another fuzzy set B. AB (x) = 1; if A (x) < B (x) A (x) T B (x) ; otherwise (1) AB (x) = 1; if A (x) B (x) A (x) T B (x) ; otherwise (2) 2.2 Implication Consider two fuzzy sets: Ain Xand Bin Y. Method1: A! B (x;y) =min[ A (x) ; B (y)] Method1: A! B (x;y) =min[1; f 1 A (x) + B (y) g] (4) 2.3 Definitions Height of a Fuzzy Set The maximum height of its membership function. hgt (A) =sup A (x) (5) Support Set Crisp set containing all elements whose membership grade is greater than 0. S=fx 2Xj A (x) >0g (6) -cutofaFuzzy Set Crisp set formed by those elements whose membership grade is greater than or equal to the threshold specified by. A=fx 2Xj A (x) g; 2[0; 1] (7) Representation Theorem A fuzzy set can be recomposed or represented, as the largest-cuts of the set for allx. A (x) =sup[ A (x)] (8) This providesamethod for reconstructinga fuzzy set using a crisp set, although sequences of crisp sets are not equal to fuzzy sets because the value of is itselfa membership grade. Also called resolution identity because the fuzzy set can be resolved into an infinite set of-cuts. Supremum The maximum of a function over a continuous inter- val. 2.4 Fuzziness and Fuzzy Resolution Fuzzy Resolution Considerf COLD, MEDIUM, HOT gagainst fCOLD, COOL, MEDIUM, HOT, SCALDINGg. The latter has a higher resolution. Degree of Fuzziness is the distance from where the membership function is 0.5 to 1, or 0.

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